Table Of ContentBoundary spectral behaviour for semiclassical
operators in one dimension
Michael Hitrik
4
Department of Mathematics, University of California, Los Angeles, CA 90095-1555,
0
0 USA
2
E-mail: [email protected]
n
a Key words: spectrum, pseudospectrum, non-selfadjoint, Hamilton map
J
2 1991/2000 AMS Subject Classification: 34E05, 34L20, 47A12
2
]
P
Abstract: For a class of non-selfadjoint semiclassical operators in dimension one,
S
. we get a complete asymptotic description of all eigenvalues near a critical value of
h
t the leading symbol of the operator on the boundary of the pseudospectrum.
a
m
[
1 Introduction and main result
1
v
8 In his work [1], E. B. Davies considers the non-selfadjoint harmonic oscillator,
8
2
P(h) = h2D2 +eiαx2, 0 < α < π/2.
1 x
0
4 By an analytic continuation argument, it was established that the spectrum of P(h)
0
consists of the eigenvalues eiα/2h(2k + 1), k = 0,1,2,... , while the semiclassical
/
h { }
pseudospectrum of P(h), defined as the range of the symbol of P(h), p(x,ξ) =
t
a ξ2 + eiαx2, on R2, is equal to the sector z C;0 argz α . As h 0,
m
{ ∈ ≤ ≤ } →
the spectrum accumulates precisely at the origin, which is the only point z at the
: 0
v boundary of the pseudospectrum, for which dp = 0 on p−1(z ) R2. The angle at
i 0 ∩
X which the eigenvalues approach the origin is given here by α/2. Now associated with
r the quadratic form p, there is a Hamilton map F, defined by
a
1 p′′ p′′ 0 1
F = xξ ξξ = ,
2 p′′ p′′ eiα 0
(cid:18) − xx − xξ (cid:19) (cid:18) − (cid:19)
with the eigenvalues µ, where µ = ieiα/2. We may therefore reformulate the result
±
of [1] by saying that the angle at which the spectrum approaches the boundary point
0 is given by arg(µ/i). In this paper we shall give a straightforward extension of
1
this result to what we think is a natural class of non-selfadjoint h-pseudodifferential
operators in dimension one. We shall show that, in the semiclassical limit, the di-
rection of the spectrum near a critical value on the boundary of the pseudospectrum
is determined by a suitable eigenvalue of the Hamilton map of the Hessian of the
principal symbol at a corresponding critical point.
Without any restriction onthe dimension, pseudospectra ofnon-selfadjoint semi-
classicalpseudodifferential operatorshavebeenstudiedintherecent paper[3],which
extended some of the results of [2], [15], [16]. Following [16] and [3], let us consider
the convection-diffusion operator with a quadratic potential,
1
P(x,hD ) = (hD )2 +i(hD )+x2 = ex/2h (hD )2 +x2 + e−x/2h.
x x x x
◦ 4 ◦
(cid:18) (cid:19)
Using the last representation, one can prove that the spectrum of P(x,hD ) is given
x
by (2k +1)h+ 1, k = 0,1,2... , while the range of the symbol of P(x,hD ) fills
{ 4 } x
out the region z C;Rez (Imz)2 . In this case, the spectrum lies strictly inside
{ ∈ ≥ }
the pseudospectrum, and Theorem 3 in [3] gives general conditions on a point z at
0
the boundary of the pseudospectrum of an operator P = pw(x,hD )+ (h), to be
x
O
away from the spectrum, as h 0. The first one is the principal type condition,
→
dp = 0 along p−1(z ), (1.1)
0
6
and the second is the following dynamical hypothesis,
For someλ C, no complete trajectory ofH is contained inp−1(z ). (1.2)
Re(λp) 0
∈
Asssuming (1.1) and (1.2) together with a basic analyticity assumption, it is estab-
lished in [3] that there is a sufficiently small but fixed neighborhood of z in C which
0
does not intersect the spectrum of P. Back to the one-dimensional case, the purpose
of this note is to describe the eigenvalue distribution near a boundary point z , in a
0
simplest situation when the condition (1.1) fails to hold (so that (1.2) fails as well).
In doing so, as in [3], we shall work under the analyticity assumptions, which will
be essential for our methods.
Let
P = Pw(x,hD ;h)
x
be the h–Weyl quantization of a symbol P(x,ξ;h), which is a holomorphic function
of (x,ξ) in a tubular neighborhood of R2 C2, with
⊂
P(x,ξ;h) = (1)m(Re(x,ξ)) (1.3)
O
2
there. Here we assume that 1 m C∞(R2) is an admissible weight function in
≤ ∈
the sense that for some fixed C ,N > 0 we have
0
m(X) C X Y Nm(Y), X,Y R2.
0
≤ h − i ∈
We next assume that as h 0, P(x,ξ;h) has an expansion in the space of holomor-
→
phic functions satisfying the bound (1.3),
∞
P(x,ξ;h) hjp (x,ξ).
j
∼
j=0
X
For h > 0 small enough, and when equipped with the domain H(m), the naturally
definedSobolevspaceassociatedtotheweightm,P becomesacloseddenselydefined
operator on L2(R).
We shall denote the principal symbol of P by p := p , and make an additional
0
assumption of the ellipticity of p near infinity,
m(ReX) 1
p(X) , X C, ImX , C > 0. (1.4)
| | ≥ C | | ≥ | | ≤ C
Throughout this paper, we shall work under the assumption that
m(X) , X , X R2. (1.5)
→ ∞ | | → ∞ ∈
As before, associated to P, we introduce the semiclassical pseudospectrum
Σ(p) := p(R2). (1.6)
This is a closed set, since p is proper, and we shall assume that Σ(p) is not all of C.
It follows then from (1.5) that if z is in the complement of Σ(p), then (P z )−1
0 0
−
exists and is a compact operator on L2. The analytic Fredholm theory implies that
the spectrum of P is discrete and consists of eigenvalues of finite multiplicity.
Let z ∂Σ(p) and assume that p−1(z ) R2 is a finite collection of points. It
0 0
∈ ∩
will in fact be sufficient to treat the case when the pre-image of z is just a single
0
point, say (0,0) R2, and in what follows we shall work under this assumption.
∈
We introduce an exterior cone condition,
There exist ǫ > 0 and θ R such that (1.7)
0 0
∈
z +(0,ǫ )ei(θ0−ǫ0,θ0+ǫ0) Σ(p) = .
0 0
∩ ∅
Our final assumptio(cid:0)n is that p z vanish(cid:1)es at (0,0) precisely to the second
0
−
order, so that in a neighborhood of this point, we have
p(X) z X 2.
0
| − | ∼ | |
The following is the main result of this work.
3
Theorem 1.1 Assume that z is at the boundary of the semiclassical pseudospec-
0
trum of P, defined in (1.6), and that p−1(z ) R2 = (0,0), with p z vanishing
0 0
∩ −
there precisely to the second order. Furthermore, we make an exterior cone assump-
tion (1.7). Then there exists α C, α = 1, such that Reαp′′(0,0) > 0, and the
∈ | |
spectrum of P in a sufficiently small but fixed neighborhood of z in C is given by
0
1
z = z +G h(k + );h + (h∞), k N.
k 0
2 O ∈
(cid:18) (cid:19)
Here G(q;h) is holomorphic in q neigh(0,C), and has an asymptotic expansion in
∈
the space of such functions, as h 0,
→
∞
G(q;h) hjG (q).
j
∼
j=0
X
We have G (0) = 0 and
0
∂ µ
arg G (0) = arg ,
0
∂q i
(cid:18) (cid:19)
(cid:16) (cid:17)
where µ is the unique eigenvalue of the Hamilton map of p′′(0,0), for which
Re(αµ/i) > 0.
We illustrate Theorem 1.1 by comparing the results of a numerical computation
of small eigenvalues of the operator P = (hD )2 +cx2 +x4, when c = 1+3i, with
x
the direction of arg(c)/2, given by the quadratic part of the symbol—see Figure 1
on the next page.
Acknowledgement. I am grateful to Maciej Zworski for fruitful discussions on the
subject of the present note. I would also like to thank Caroline Lasser for helpful
discussions and advice, and Johannes Sj¨ostrand for stimulating comments. The
partial support of the National Science Foundation under grant DMS-0304970 is
gratefully acknowledged.
2 Proof of Theorem 1.1
Throughout this section, we shall assume, as we may, that z = 0. Our starting
0
point is the following essentially well-known result.
4
0.08
h=0.01
h=0.001
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Figure 1: Numerical computation of the eigenvalues of P = (hD )2+cx2+x4, when
x
c = 1 + 3i, for two different values of h. When computing the eigenvalues of P,
following [14], we discretized the operator using the Chebyshev spectral method.
To compare the results of the computation with Theorem 1.1 we have also plotted
the black solid line of the slope tan(arg(c)/2) 0.7208.
≈
5
Proposition 2.1 Let q(x,ξ) be a complex-valued quadratic form on R2 such that
q(x,ξ) = 0 precisely when (x,ξ) = (0,0). Then the range of q on R2 is either all of
C or there exists a proper closed convex cone C which contains the range of q.
⊂
Proof: This result is established in Lemma 3.1 of [10], and we shall recall the proof
for completeness only. In suitable linear coordinates on R2, we may write for some
c = 0,
6
q(x,ξ) = c(x iξ)(x λξ), Imλ = 0.
− − 6
Writing z = x+iξ, we find that in the case when Imλ < 0,
1 2i
q = z(z +γz), α = ,
α c(i λ)
−
where γ = (i+λ)(i λ)−1 satisfies γ < 1. Therefore,
− | |
(z)2
Re(αq) = Re( z 2 +γ(z)2) = Re z 2 1+γ > 0,
| | | | z 2
(cid:18) (cid:18) | | (cid:19)(cid:19)
when z = 0, and in this case the range of q is contained in a closed angle of opening
6
< π. In the opposite case, Imλ > 0, we find that with some β = 0,
6
βq = γ z 2 +(z)2, γ < 1.
| | | |
It follows that the argument variation of βq along the circle z = 1 is non-vanishing,
| |
and hence the range of q on R2 is all of C. 2
Remark. According to Theorem 2.1.18 in [6], the range of a general complex-valued
quadratic form on R2, restricted to z = 1, is an ellipse, possibly degenerated to an
| |
interval or a point.
Applying Proposition 2.1 to the quadratic form p (X) = p′′(0,0)X,X /2, which
2
h i
begins the Taylor expansion of p at the origin, and using the exterior cone condition
(1.7), we conclude that since 0 is a boundary point of the pseudospectrum of P,
the range of p is not all of C, for otherwise, the argument variation of p along
2
the positively oriented boundary of every small disc centered at 0, would be non-
vanishing. We then know that for some α C, α = 1, the real part of αp
2
∈ | |
is positive definite. After a multiplication of P by α, we may and will assume
henceforth that α = 1, so that the range of p on R2 is contained in a set of the
2
form
Γ = w C; Imw CRew ,
{ ∈ | | ≤ }
for some C > 0.
6
By polarizaton, p (X) gives rise to a symmetric bilinear form on C2, and then
2
we write
p (X,Y) = σ(X,FY), X,Y C2,
2
∈
where σ is the complex symplectic form on C2, and F : C2 C2 is the Hamilton
→
mapofp —seeSection21.5of[5]andalso[7],forasystematicdiscussionofquadratic
2
forms on symplectic vector spaces. Since Rep is positive definite on R2, it is true
2
that F is bijective, and it is well known and easily seen that if µ is an eigenvalue of
F then so is µ. When X is an eigenvector of F corresponding to the eigenvalue µ,
−
we write
p (X,X) = σ(X,µX) = 2iµσ(ReX,ImX).
2
Here the left hand side is equal to p (ReX,ReX) + p (ImX,ImX), and then we
2 2
see that either µ/i Γ or µ/i Γ. In what follows we shall let µ stand for the
∈ − ∈
unique eigenvalue of F for which Re(µ/i) > 0. Our goal is to show that as h 0,
→
the spectrum of P near 0 accumulates towards the origin in the direction given by
the argument of µ/i.
Remark. In Theorem 3.5 of [10], the spectrum of the Weyl quantization of a general
complex valued elliptic quadratic form was computed. When specialized to the case
at hand, the results of [10] show that the spectrum of pw(x,hD) is of the form
2
iµh(2k + 1), k = 0,1,2,... . Our Theorem 1.1 should therefore be viewed as
{− }
an extension of the one-dimensional version of Theorem 3.5 of [10] to the case of
a general analytic h-pseudodifferential operator. The proof of Theorem 3.5 in [10]
employs complex linear canonical transformations, in order to essentially reduce the
quadraticformgloballytoacomplexmultipleoftheharmonicoscillator. Inourcase,
we shall proceed similarly, but now merely microlocally, working near the critical
point (0,0) of the leading symbol of P.
We shall first use real canonical transformations to simplify p . After a real linear
2
symplectic change of coordinates, and a conjugation of P by means of a correspon-
ding unitary metaplectic operator, we may assume that as (x,ξ) (0,0),
→
λ i
p(x,ξ) = p (x,ξ)+ ((x,ξ)3) = (x2+ξ2)+ (ax2+bξ2+2cxξ)+ ((x,ξ)3), λ > 0,
2
O 2 2 O
(2.1)
for some a,b,c R. Here the quadratic form Imp can be diagonalized by means
2
∈
of an orthogonal transformation, which may be assumed to be orientation preser-
ving. Hence this transformation is symplectic, since we are in dimension one, and
we conclude that after an additional real linear symplectic transformation and a
7
metaplectic conjugation, we may reduce the quadratic part of p to the form in (2.1)
with c = 0. We then write
(λ+ia)
p(x,ξ) = x2 +dξ2 + ((x,ξ)3),
2 O
(cid:0) (cid:1)
where d = (λ+ia)−1(λ+ib), and it is no restriction to assume that Imd 0. After
≥
an additional real symplectic dilation in (x,ξ), the symbol p(x,ξ) takes the form
(λ+ia) d 1/2
p(x,ξ) = | | x2 +eiαξ2 + ((x,ξ)3), α = argd [0,π). (2.2)
2 O ∈
(cid:0) (cid:1)
Assuming thatα > 0, we shall now lookfora small IR-deformationΛ ofR2, such
that the restriction of the quadratic part of p to Λ will have a constant argument,
in a neighborhood of (0,0). We first argue formally, and consider Λ = T∗(eiθR),
0
where θ (0,π/4) is to be chosen. Notice that with G(x,ξ) = θxξ, we can write
∈
Λ = exp(iH )(R2). If (x,ξ) Λ then x = eiθy, ξ = e−iθη for (y,η) R2, and
0 G 0
∈ ∈
using (y,η) as real symplectic coordinates on Λ , we get
0
(λ+ia) d 1/2
p = p(exp(iH )(y,η)) = | | e2iθy2 +eiαe−2iθη2 + ((y,η)3).
|Λ0 G 2 O
(cid:0) (cid:1)
We see that if we choose θ = α/4, then
(λ+ia)d1/2
p(exp(iH )(y,η)) = y2 +η2 + ((y,η)3)
G
2 O
((λ+ia)(λ+(cid:0)ib))1/2 (cid:1)
= (y2+η2)+ ((y,η)3).
2 O
Here we take the square root with the positive real part. Using the symplectic
invariance of the Hamilton map F of p , we then check that here the coefficient
2
((λ+ia)(λ+ib))1/2/2
is equal to µ/i, where µ is the eigenvalue of F, for which Re(µ/i) > 0.
We shall now introduce a global IR-manifold Λ which in a very small complex
neighborhood of (0,0) agrees with T∗(eiθR), and further away from that set it is
equal to R2. The manifold Λ will also be chosen so that the restriction of p to
Λ (0,0) is non-vanishing. We remark that the following construction of Λ is
\{ }
similar to some arguments of [8] and [12] in the theory of resonances.
It will be convenient to work with the following FBI–Bargmann transform,
Tu(x) = Ch−3/4 eiϕ(x,y)/hu(y)dy, x C, C > 0, (2.3)
∈
Z
8
where
i x2 y2
ϕ(x,y) = + √2yx .
eiα/2 2 2 −
(cid:18) (cid:19)
Notice that Imϕ′′ > 0 and ϕ′′ = 0, so that ϕ(x,y) is an admissible phase in
yy xy 6
(2.3). Associated to T, there is a complex linear canonical transformation κ :
T
(y, ϕ′(x,y)) (x,ϕ (x,y)), given by
− y 7→ x
1
κ (y,η) = y ieiα/2η, ie−iα/2y +η , (y,η) C2.
T
√2 − − ∈
(cid:0) (cid:1)
From the general theory [11], we know that κ maps R2 bijectively onto
T
2
Λ := x, ∂ Φ (x) , x C ,
Φ0 i x 0 ∈
(cid:26)(cid:18) (cid:19) (cid:27)
where Φ is the strictly subharmonic quadratic form given by
0
Φ (x) = sup Imϕ(x,y).
0
y∈R−
A straightforward computation shows that
(Rex)2 1+sin2(α/2)(Imx)2
Φ (x) = cos(α/2) + +sin(α/2)RexImx. (2.4)
0
2 cos(α/2) 2
We remark that Φ is even strictly convex. Another computation shows next that
0
the image of T∗(eiθR), θ = α/4, under κ is of the form Λ , where Φ (x) = 1 x 2.
T Φ1 1 2 | |
When ǫ > 0, we take χ C∞(C), 0 χ 1, such that χ = 1 when x < e1/ǫ,
ǫ ∈ 0 ≤ ≤ ǫ | |
χ = 0 for x > e2/ǫ, and such that
ǫ
| |
∂αχ (x) = (ǫ) x −|α|, α > 0.
ǫ α
O | | | |
Then the smooth function
Φ = χ Φ +(1 χ )Φ
ǫ 1 ǫ 0
−
is strictly convex for ǫ > 0 smaell enough, and satisfies
Φ(x) = Φ (x), x e1/ǫ,
1
| | ≤
Φe(x) = Φ (x), x e2/ǫ.
0
| | ≥
e
9
In order to decrease the size of the neighborhood of 0 where Φ = Φ , we introduce
0
6
the strictly convex function
e
2
η K x
Φ (x) = Φ ǫ , K := e2/ǫ, 0 < η 1, (2.5)
η ǫ
K η ≪
(cid:18) ǫ(cid:19) (cid:18) (cid:19)
e
so that Φ (x) = Φ (x) for x η. Notice that taking η sufficiently small, we
η 0
| | ≥
may achieve that Φ Φ is arbitrarily small in C1–norm. We then introduce the
η 0
−
IR-manifold
2
Λ : ξ = ∂ Φ ,
Φη i x η
and remark that along Λ , we have
Φη
K x 2∂Φ (x) K x 2∂Φ (x)
ǫ 1 ǫ 0
ξ = ξ(x) = χ + 1 χ + (ǫ x ),
ǫ ǫ
η i ∂x − η i ∂x O | |
(cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
uniformly in η. We claim now that with p = p κ−1,
◦ T
1
p(x,ξ(x) e x 2, C > 0, (2.6)
| | ≥ C | |
if x neigh(0,C) and ǫ > 0eis small enough. Indeeed, using (2.2) and computing the
∈ e
inverse of κ , we find that
T
p(x,ξ) = 2((λ+ia)(λ+ib))1/2ixξ + ((x,ξ)3).
O
Using the strict convexity of the quadratic forms Φ and Φ , we get
e 0 1
Re(ixξ(x)) = χ x 2 +(1 χ )Re(2x∂ Φ )+ (ǫ x 2)
ǫ ǫ x 0
| | − O | |
= χ x 2 +(1 χ ) x, Φ (x) + (ǫ x 2) x 2,
ǫ ǫ 0
| | − h ∇ i O | | ∼ | |
uniformly in ǫ and η, and the transversal ellipticity property (2.6) follows at once.
We conclude that for every fixed small η > 0, the IR-manifold Λ := (κ )−1(Λ ) is
T Φη
equal to R2 away from an arbitrarily small, previously given neighborhood of (0,0)
and it agrees with T∗(eiθR) near (0,0). Moreover, it is clear that the restriction of
p to Λ vanishes precisely at (0,0).
Let us now recall the h-dependent Hilbert space H(Λ) L2(R), associated with
≃
the IR-manifold Λ and equipped with the norm obtained by replacing the weight
Φ by Φ = Φ on the FBI transform side. We also introduce the Sobolev space
0 η
H(Λ,m) which agrees with H(m) as a set, and whose norm is obtained from the
10
